jmc

First, solve the inequality so that only the absolute value quantity is on the left and the constant value is on the right.

$$\begin{array}{rl}2|x|+7& <17\\ 2|x|& <10\\ |x|& <5\end{array}$$To solve the inequality which has an absolute value in it, we must turn this into two different inequalities, one as normal, one with a reversed sign and opposite resulting value. Both will have the absolute value removed.

$$\begin{array}{rl}x& <5\\ x& >-5\end{array}$$Since we need the least integer value of $x$, and $x$ has to be $\text{greater than }$ -5, the next smallest integer is $\boxed{-4}$.